 \(K_{n}/n \rightarrow 1\)  and \(n-K_{n} \rightarrow \infty \) , \(F\)  is \(\Delta \) -analytic and there exist \(\alpha , c>0\)  such that \(F(\rho -z) \sim c/z^{\alpha }\)  as \(z \rightarrow 0\)  with \(\mathrm {Re}(z)>0\) , and finally \(v_{n} = \frac{4(1+\alpha )n^{2}}{9\alpha (n-K_{n})}\) .
As for random paths, in the first regime, since \(F^{(2)}(b) < \infty \) , then essentially the degree of a typical face in \(M_{n,K_n}\)  has uniformly bounded variance and the Brownian map is naturally expected in the limit.
In the second regime, we force the number of vertices to be small compared to the number of edges. Clearly, this requires \(k_{0}=1\) , i.e. \(q_{1} > 0\) : in order to have much less vertices than edges, the map must have (many!) double edges, i.e. faces with degree 2. On the other hand we also need the extra assumption \(q_{2} > 0\)  to avoid periodicity issues which would require a more technical analysis. However, there is no additional regularity assumption on \(F\) .
Finally, in the last regime, when the number of vertices is asymptotically equivalent to the number of edges, we recall that \(F\)  is \( \Delta \) -analytic if there exist \(R>\rho \)  and \(\phi \in (0,\pi /2)\)  such that \(F\)  is analytic on \( \lbrace z : |z|<R, z \ne \rho , |\arg (z-1)|>\phi \rbrace \) , see [1]}. This condition on \(F\)  is crucial and Theorem REF  below shows a very different behaviour.
Our final result concerning random maps focuses on the case \(K_{n}/n \rightarrow A(\rho )\)  when the latter is finite. Here, we assume that the probability distribution with generating function \(F(\rho \,\cdot )/F(\rho )\)  is aperiodic and belongs to the domain of attraction of a stable law with index \(\alpha \in (1, 2]\) , i.e. that there exist \(\alpha \in (1,2]\) , \(\rho >0\) , \(m>0\) , and a slowly varying function \(L\)  such that for every \(0 \leqslant s < \rho \) ,
\(F(s)=F(\rho ) \mathopen {}\mathclose {\left(1-m+\frac{m s}{\rho }+\mathopen {}\mathclose {\left(1-\frac{s}{\rho }\right)}^{\alpha } L\mathopen {}\mathclose {\left(\frac{1}{1-s/\rho }\right)}\right)}.\) 
In the case \(F^{(2)}(\rho ) < \infty \) , i.e. when \(\alpha =2\)  and \(L\)  admits a finite limit at infinity, say \(\ell \) , the law with generating function \(F(\rho \,\cdot )/F(\rho )\)  has a finite variance, equal to
\(\sigma ^{2} = 2\ell + m - m^{2}= \frac{\rho ^{2} F^{(2)}(\rho )}{F(\rho )} + \frac{\rho F^{\prime }(\rho )}{F(\rho )} - \mathopen {}\mathclose {\left(\frac{\rho F^{\prime }(\rho )}{F(\rho )}\right)}^{2}.\) 
In this case we set \(r_{n} = \sqrt{n \sigma ^{2}/2}\)  for every \(n\geqslant 1\) , otherwise, we let \((r_{n})_{n \geqslant 1}\)  be a sequence such that \(r_{n}^{\alpha } \sim n L(r_{n})\)  as \(n \rightarrow \infty \) .
Example 6.3 Let us give an explicit example taken from [2]}. The weights
\(q_k = \frac{-\sqrt{\pi }}{2\Gamma (1-\alpha )} \mathopen {}\mathclose {\left(\frac{1}{4\alpha }\right)}^{k-1} \frac{\Gamma (-\alpha +k)}{\Gamma (\frac{1}{2}+k)}\qquad \text{for}k \geqslant 2.\) 
give rise to the generation function \(F\)  satisfying \(F(\alpha z)/F(\alpha ) = z + \alpha ^{-1} (1-z)^\alpha \) . In this case, \(\rho =\alpha \) , \(m=1\) , \(F(\alpha )=\alpha \) , \(F^{\prime }(\rho )=1\) , \(F^{(2)}(\rho )=1/2\)  when \(\alpha =2\) , \(L\)  is constant equal to 1, and finally \(A(\alpha ) = 1/\alpha \) . In particular, one can take \(r_{n}=n^{1/\alpha }\) .

The next statements are our last applications to random maps.
Theorem 6.4 
Under the previous assumptions, assume that \(K_{n}\)  takes the form
\(K_n = A(\rho ) n + \lambda _{n},\qquad \text{where}\qquad \frac{\lambda _{n}}{r_{n}} \mathop {\longrightarrow }^{}_{n \rightarrow \infty }\lambda \in [-\infty ,\infty ].\) 
The following convergences hold for the Gromov–Hausdorff–Prokhorov topology:

If \(\lambda \in \mathbb {R}\)  and \(\alpha = 2\) , and
\(v_{n} = \frac{\rho F^{(2)}(\rho )}{F^{\prime }(\rho )} \frac{4n}{9} \mathbb {1}_{\lbrace F^{(2)}(\rho ) < \infty \rbrace }+ \frac{F(\rho )}{\rho F^{\prime }(\rho )}\frac{8 r_{n}^{2}}{9} \mathbb {1}_{\lbrace F^{(2)}(\rho ) = \infty \rbrace }.\) 
then
\(\mathopen {}\mathclose {\left(V(M_{n,K_n}), v_{n}^{-1/4} d_{\mathrm {gr}}, p_{\mathrm {unif}}\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }({M}, {D}, {m}),\) 
where \(({M}, {D}, {m})\)  is the Brownian map.

 If \(\lambda = -\infty \)  and \(\lambda _{n}/n \rightarrow 0\) , let \(\varepsilon _{n} \rightarrow 0\)  be defined by \(A(\rho (1-\varepsilon _{n})) = K_{n}/n\) , then for
\(v_{n} = \frac{\rho F^{(2)}(\rho )}{F^{\prime }(\rho )} \frac{4n}{9} \mathbb {1}_{\lbrace F^{(2)}(\rho ) < \infty \rbrace }+ \frac{4}{9} \frac{\alpha -1}{1-A(\rho )} \frac{|\lambda _{n}|}{\varepsilon _{n}} \mathbb {1}_{\lbrace F^{(2)}(\rho ) = \infty \rbrace },\) 
we have
\(\mathopen {}\mathclose {\left(V(M_{n,K_n}), v_{n}^{-1/4} d_{\mathrm {gr}}, p_{\mathrm {unif}}\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }({M}, {D}, {m}),\) 
where \(({M}, {D}, {m})\)  is the Brownian map.

 If \(\lambda = \infty \) , we further assume that \(\binom{2k-1}{k-1} q_{k} \rho ^{-k}\)  is regularly varying at infinity with some index \(-\beta < -2\) .
If \(F^{(2)}(\rho )<\infty \)  (which implies \(\beta \geqslant 3\) ), we assume furthermore that there exists \(c>(\beta -3)/\sigma ^{2}\)  such that \(\lambda _{n} \geqslant \sqrt{c \ln (n)}\) .
Then
\(\mathopen {}\mathclose {\left(V(M_{n,K_n}), (2\lambda _{n})^{-1/2} d_{\mathrm {gr}}, p_{\mathrm {unif}}\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }({T}, {d}, {p})\) 
where \(({T}, {d}, {p})\)  is Aldous' Brownian tree coded by the standard Brownian excursion.
We shall establish these convergences by combining Theorem REF  with [3]}. Our last result concerns the first regime, \(\lambda \in \mathbb {R}\) , when \(\alpha \in (1,2)\) . In this case, we expect new limit spaces to appear, and in this direction here we establish a tightness result as a first step.
Theorem 6.5 
In the same framework as the previous theorem, still assuming that \(K_{n}\)  takes the form
\(K_n = A(\rho ) n + \lambda _{n},\qquad \text{where}\qquad \frac{\lambda _{n}}{r_{n}} \mathop {\longrightarrow }^{}_{n \rightarrow \infty }\lambda \in \mathbb {R}.\) 
When \(\alpha \in (1,2)\) , the sequence
\(\mathopen {}\mathclose {\left(V(M_{n,K_n}), \frac{1}{\sqrt{r_{n}}} d_{\mathrm {gr}}, p_{\mathrm {unif}}\right)}_{n \geqslant 1}\) 
is tight for the Gromov–Hausdorff–Prokhorov topology and nondegenerate (in the sense that it does not converge towards a deterministic limit).

In the case of maps solely conditioned by their number of vertices, Le Gall & Miermont [4]} proved such a tightness result by showing that the label process
(which records the label of the vertices of the tree in depth-first search order) converges in distribution. This enabled them to show that all subsequential limits have the same Hausdorff dimension \(2\alpha \)  almost surely; furthermore some limit theorems then do not require the extraction of a subsequence, such as the convergence of the profile of distances to an independent uniform random vertex.
Here, Theorem REF  follows by a direct application of [3]} which only shows tightness of the label process, which does not provide any extra information on the limit spaces other than the scale of the typical distances.
In the companion paper [6]}, we extend Le Gall & Miermont's results which involved the so-called `continuous distance process' based on the excursion of stable Lévy process to more general Lévy processes with no negative jump.
Applied to the present setting of biconditioned maps, we obtain that all subsequential limits in Theorem REF  have the same Hausdorff, packing, and Minkowski dimensions, all equal to \(2\alpha \)  almost surely, independently of \(\lambda \in \mathbb {R}\) .
Finally, in the case \(\lambda =0\) , the limit spaces are the same as in [4]}.
Remark 6.6 It is interesting to note that in the case \(m<1\) , under an additional regularity assumption, the scaling limit of \({q}\) -Boltzmann planar maps conditioned on having a large number of edges is the Brownian CRT (see [8]} and [3]}), but an appropriate biconditioning also allows to escape this universality class.

[Proof of Theorems REF , REF , and REF ]
The paper [3]} specifically studies a model of uniform random maps `with prescribed degrees' in the sense that given, say, \(N\)  positive integers \(d_{N,1} \geqslant \cdots \geqslant d_{N,N}\) , a (rooted planar) map is sampled uniformly at random amongst all those with \(N\)  faces, with degrees \(2d_{N,1}, \dots , 2d_{N,N}\) . Then one wants to understand to behaviour of such a map \(M_{N}\)  as \(N \rightarrow \infty \)  in terms of the triangular array \((d_{N,i})_{N \geqslant i \geqslant 1}\) . It is shown in [3]} that
the sequence
\(\mathopen {}\mathclose {\left(V(M_{N}), \sigma _{N}^{-1/2} d_{\mathrm {gr}}, p_{\mathrm {unif}}\right)}_{n \geqslant 1}\qquad \text{where}\qquad \sigma _{N}^{2} = \sum _{i=1}^{N} d_{N,i} (d_{N,i}-1),\) 
is always tight for the Gromov–Hausdorff–Prokhorov topology.
Moreover, by Theorem 2 and Theorem 3 respectively in [3]}, as \(N \rightarrow \infty \) ,

if \(\sigma _{N}^{-1} d_{N,1} \rightarrow 0\) , then it converges in distribution towards \(\sqrt{2/3}\)  times the Brownian map \(({M}, {D}, {m})\) ;

if \(\sigma _{N}^{-1} d_{N,1} \rightarrow 1\) ,
then it converges in distribution towards \(\sqrt{2}\)  times the Brownian tree \(({T}, {d}, {p})\) .

Since two maps with same face degrees have the same \(w^{{q}}\) -weight, a size-conditioned Boltzmann map \(M_{n,K_n}\)  sampled from \(\mathbb {P}^{{q}}_{n,K_{n}}\)  can be seen as a mixture of this model with \(N=n-K_{n}\) : One first samples the degrees \(d_{n-K_{n},1} \geqslant \cdots \geqslant d_{n-K_{n},n-K_{n}}\)  randomly with an appropriate distribution, and then conditionally given this sequence, the map \(M_{n,K_n}\)  has the uniform distribution on maps with \(n-K_{n}\)  faces, with degrees respectively \(2d_{n-K_{n},1}, \dots , 2d_{n-K_{n},n-K_{n}}\) .
We then deduce that, once rescaled by a, now random, factor, the sequence of maps \((\sigma _{n-K_{n}}^{-1/2} M_{n,K_n})_{n \geqslant 1}\)  is tight, and it suffices to prove the convergence in probability \(d_{n-K_{n},1} / \sigma _{n-K_{n}} \rightarrow 0\)  as \(n \rightarrow \infty \)  to deduce that they converge to \(\sqrt{2/3}\)  times the Brownian map, or that \(d_{n-K_{n},1} / \sigma _{n-K_{n}} \rightarrow 1\)  in probability to deduce that they converge to \(\sqrt{2}\)  times the Brownian tree.
According to Sec. REF , the half face-degrees of \(M_{n,K_{n}}\)  are given by the nonnegative increments plus one of the Łukasiewicz path of the associated tree. Further, from Sec. REF , the latter has the law \(\mathbb {P}^{{\theta },+}_{n,K_{n}}\) , where \({\theta }\)  is defined by (REF ). If we denote by \(X^{n}_{k}\)  the increments of this path, then the quantity \(\sigma _{n-K_{n}}^{2}\)  has therefore the same law as
\(\sum _{k=1}^{n} X^{n}_{k} (X^{n}_{k}+1) = -1 + \sum _{k=1}^{n} (X^{n}_{k})^{2}.\) 
We infer from Theorems REF  and REF , with also Remark REF  in the finite variance regimes that:

The random factor \(\sigma _{n-K_{n}}\)  can be replaced by the deterministic factor used in Theorems REF , Theorem REF , and REF  respectively;

The ratio \(d_{n-K_{n},1} / \sigma _{n-K_{n}}\)  converges in probability to 0 as \(n \rightarrow \infty \)  in Theorem REF  as well as in Theorem REF  and in Theorem REF .

The ratio \(d_{n-K_{n},1} / \sigma _{n-K_{n}}\)  converges in probability to 1 as \(n \rightarrow \infty \)  in Theorem REF .

Our claim then follows from [3]}.
On the sum of the increments squared
In this section, we provide the proof of the convergence of the sum of the squares of the increments of the simply generated random path \(S^{n}\)  conditioned to end at \(S^{n}_{n} = x_{n}\) , as stated in
Theorems REF , REF ,
and finally REF  in the case \(\alpha =2\) . These results were needed when dealing with random maps just above.
Throughout this section, we shall denote by \(X^{n}_{k} = S^{n}_{k}-S^{n}_{k-1}\)  for every \(1 \leqslant k \leqslant n\)  the increments of the random bridge \(S^{n}\) .
Each regime requires a different argument and is studied in a separate subsection.
Recall the notation \(G\)  and \(\Psi \)  from (REF ).
In each regime studied from Sec. REF  to Sec. REF , we have \(x_{n}/n < \Psi (\rho )\)  for every \(n\)  large enough so we can define as in the proof of Proposition REF  a random variable \(\widehat{X}^{n}\)  with generating function \(\widehat{G}^{n} = G(b_{n} \,\cdot ) / G(b_{n})\) , where \(b_{n}\)  is such that \(\mathbb {E}[\widehat{X}^{n}] = \Psi (b_{n}) = x_{n}/n\) . We will use the explicit expressions of the first moments of \(\widehat{X}^{n}\) , which are given by (REF ).
We let further \((\widehat{S}^{n}_{i} ; i \geqslant 0)\)  denote a random walk with such i.i.d. steps.
On the other hand, in Sec. REF , we have \(x_{n}/n \geqslant \Psi (\rho )\) , so the latter is finite and we now define \(\widehat{X}\)  similarly but with \(b = \rho \)  instead of \(b_{n}\) ; then \((\widehat{S}_{i} ; i \geqslant 0)\)  shall denote a random walk with such i.i.d. steps.
In both cases, Lemma REF  shows that the paths \((\widehat{S}^{n}_{i} ; 0 \leqslant i \leqslant n)\)  under \(\mathbb {P}(\,\cdot \mid \widehat{S}^{n}_{n} = x_{n})\)  and \((\widehat{S}_{i} ; 0 \leqslant i \leqslant n)\)  under \(\mathbb {P}(\,\cdot \mid \widehat{S}_{n} = x_{n})\)  have the same law as our original path \(S^{n}\) .
The bulk regime
In this first subsection, we focus on Theorem REF , i.e. we assume that \(x_{n}/n\)  converges to some limit \(\gamma \in (i_{{p}}, \Psi (\rho ))\) .
Let \(b_{n}\)  and \(\widehat{X}^{n}\)  as above, then \(b_{n} \rightarrow b\)  where \(b \in (0, \rho )\)  is such that \(\Psi (b) = \gamma \) ; we let similarly \(\widehat{X}^{b}\)  have the generating function \(\widehat{G}^{b}(s) = G(b s) / G(b)\) .
The next result completes Theorem REF  and was used in the proof of Theorem REF .
Proposition A.1 
The following convergence holds in probability:
\(\frac{1}{n} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\mathbb {E}\big [(\widehat{X}^{b})^{2}\big ] = \frac{b^{2} G^{(2)}(b) + b G^{\prime }(b)}{G(b)}.\) 


According to Lemma REF , the law of the path \(S^{n}\)  and the random walk \(\widehat{S}^{n}\)  under \(\mathbb {P}(\,\cdot \mid \widehat{S}^{n}_{n}=x_{n})\)  are the same so we may replace the increments of the former by those of the latter. Let us therefore consider a sequence \((\widehat{X}^{n}_{k})_{k \geqslant 1}\)  of i.i.d. random variables with generating function \(\widehat{G}^{n} = G(b_{n} \,\cdot ) / G(b_{n})\) , and recall their second moment from (REF ), namely
\(s^{2}_{n} = \mathbb {E}\big [(\widehat{X}^{n}_{1})^{2}\big ]= \frac{b_{n}^{2} G^{(2)}(b_{n}) + b_{n} G^{\prime }(b_{n})}{G(b_{n})}.\) 
Then \(s^{2}_{n} \rightarrow s^{2} = \mathbb {E}[(\widehat{X}^{b})^{2}]\) .
Fix \(\varepsilon >0\) ; the proof by truncation of the weak law of large numbers provided e.g. in [14]}, shows the following: let \(\Sigma _{n} = \sum _{k=1}^{n} ((\widehat{X}^{n}_{k})^{2} - s_{n}^{2})\)  and \(\Sigma _{n}^{\prime } = \sum _{k=1}^{n} ((\widehat{X}^{n}_{k})^{2} - s_{n}^{2}) \mathbb {1}_{\lbrace |(\widehat{X}^{n}_{k})^{2} - s_{n}^{2}| \leqslant n \varepsilon ^{3}\rbrace }\) , then

P(|n-E[n']| > n )
  E[|(Xn1)2 - sn2|] + n  P(|(Xn1)2 - sn2| > n 3)
2sn2 + -3  E[|(Xn1)2 - sn2| 1{|(Xn1)2 - sn2| > n 3}],

and
\(n^{-1} \mathopen {}\mathclose {\left|\mathbb {E}\mathopen {}\mathclose {\left[\Sigma _{n}^{\prime }\right]}\right|} \leqslant \mathbb {E}\mathopen {}\mathclose {\left[\mathopen {}\mathclose {\left|(\widehat{X}^{n}_{1})^{2} - s_{n}^{2}\right|} \mathbb {1}_{\lbrace |(\widehat{X}^{n}_{1})^{2} - s_{n}^{2}| > n \varepsilon ^{3}\rbrace }\right]}.\) 
Recall the fourth moment of \(\widehat{X}^{n}_{1}\)  from (REF ); since \(b_{n} \rightarrow b \in (0,\rho )\) , then \(\mathbb {E}[(\widehat{X}^{n}_{1})^{4}]\)  converges to the similar quantity with \(b\)  in place of \(b_{n}\)  (which equals \(\mathbb {E}[(\widehat{X}^{b})^{4}]\) ). Since \(s_{n}^{2} \rightarrow s^{2}\) , we deduce that \(((\widehat{X}^{n}_{1})^{2} - s_{n}^{2})_{n}\)  is uniformly bounded in \(L^{2}\)  and therefore the right-hand side in the preceding display converges to 0. Hence,
\(\lim _{n\rightarrow \infty } n^{-1} \mathbb {E}\mathopen {}\mathclose {\left[\Sigma _{n}^{\prime }\right]} = 0\qquad \text{and}\qquad \limsup _{n\rightarrow \infty } \mathbb {P}\mathopen {}\mathclose {\left(\mathopen {}\mathclose {\left|\Sigma _{n}-\mathbb {E}\mathopen {}\mathclose {\left[\Sigma _{n}^{\prime }\right]}\right|} > n \varepsilon \right)}\leqslant 2\varepsilon s^{2}.\) 
Since this holds for every \(\varepsilon >0\) , we conclude that \(n^{-1} \Sigma _{n}\)  converges to 0 in probability, so
\(\frac{1}{n} \sum _{k=1}^{n} (\widehat{X}_{k}^{n})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }s^{2},\) 
under the unconditional law. Then as in the proof of Proposition REF , by absolute continuity, we can transfer this convergence to the first half of the bridge, i.e. under \(\mathbb {P}( \, \cdot \mid \widehat{S}^{n}_{n}=x_{n})\) ; by time-reversal the same holds also for the second half of the bridge. Since this bridge of \(\widehat{S}^{n}\)  has the same law as our path \(S^{n}\) , this shows our assertion.

The small endpoint regime
We focus in this subsection on Theorem REF : we only assume that \(p(0), p(1) > 0\)  and we suppose that \(x_{n}/n \rightarrow 0\) .
The next result completes Theorem REF  and was used in the proof of Theorem REF .
Proposition A.2 
The following convergence in probability holds:
\(\frac{1}{x_{n}} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }1.\) 


Exactly as in the proof of Proposition REF ,
it is sufficient to work under the unconditional law of the random walk \(\widehat{S}^{n}\) , and then conclude by absolute continuity and time-reversal. We therefore replace the \(X^{n}_{k}\) 's by a sequence \((\widehat{X}^{n}_{k})_{k \geqslant 1}\)  of i.i.d. random variables with generating function \(\widehat{G}^{n} = G(b_{n} \,\cdot ) / G(b_{n})\) , where \(b_{n} \in (0, \rho )\)  is such that \(\Psi (b_{n}) = x_{n} / n\) , so \(b_{n} \rightarrow 0\) .
We then derive from (REF ) the mean and variance of \((\widehat{X}^{n}_{1})^{2}\) , and further, by (REF ),
\(\mathbb {E}\big [(\widehat{X}^{n}_{1})^{2}\big ] = \frac{b_{n}^{2} G^{(2)}(b_{n}) + b_{n} G^{\prime }(b_{n})}{G(b_{n})}= \frac{x_{n}}{n} \mathopen {}\mathclose {\left(\frac{b_{n}^{2} G^{(2)}(b_{n})}{b_{n} G^{\prime }(b_{n})} + 1\right)}\mathop {\sim }^{}_{n \rightarrow \infty }\frac{x_{n}}{n},\) 
and similarly
\(\mathbb {E}\big [(\widehat{X}^{n}_{1})^{4}\big ]= \frac{x_{n}}{n} \mathopen {}\mathclose {\left(\frac{b_{n}^{4} G^{(4)}(b_{n}) + 6 b_{n}^{3} G^{(3)}(b_{n}) - 11 b_{n}^{2} G^{(2)}(b_{n})}{b_{n} G^{\prime }(b_{n})} + 6\right)},\) 
so
\(\mathrm {Var}\big ((\widehat{X}^{n}_{1})^{2}\big ) \mathop {\sim }^{}_{n \rightarrow \infty }6 \frac{x_{n}}{n}.\) 
Since \(x_{n} \rightarrow \infty \) , we infer that \(\frac{1}{x_{n}} \sum _{k=1}^{n} (X^{n}_{k})^{2}\)  converges to 1 in \(L^{2}\)  under the unconditioned probability, which suffices to conclude.

The large endpoint regime
In this subsection, we focus on Theorem REF : We now assume that \(x_{n}/n \rightarrow \infty \) , that \(G\)  is \(\Delta \) -analytic, and that there exist \(c, \rho , \alpha >0\)  such that \(G(\rho -z) \sim c z^{-\alpha }\)  as \(z \rightarrow 0\)  with \(\mathrm {Re}(z)>0\) .
The next result completes Theorem REF  and was used in the proof of Theorem REF .
Proposition A.3 
The following convergence in probability holds:
\(\frac{n}{x_{n}^{2}} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\frac{\alpha +1}{\alpha }.\) 


The proof is similar to that of Proposition REF . First,
it is sufficient to work under the unconditional law of the random walk \(\widehat{S}^{n}\) , and then conclude by absolute continuity and time-reversal that the same holds for the bridge, which has the same law as \(S^{n}\) . We therefore replace the \(X^{n}_{k}\) 's by a sequence \((\widehat{X}^{n}_{k})_{k \geqslant 1}\)  of i.i.d. random variables with generating function \(\widehat{G}^{n} = G(b_{n} \,\cdot ) / G(b_{n})\) .
Second, we rely on the moments of \((\widehat{X}^{n}_{1})^{2}\)  calculated in (REF ).
Indeed, we already noticed in (REF ) that, as \(n\rightarrow \infty \) ,
\(\mathbb {E}\big [(\widehat{X}^{n}_{1})^{2}\big ] \sim \frac{\alpha +1}{\alpha } \frac{x_{n}^{2}}{n^{2}}.\) 
Similarly, combining the expression of the moments from (REF ) and the derivatives of \(G\)  from (REF ) we deduce that as \(n\rightarrow \infty \) ,
\(\mathrm {Var}\big ((\widehat{X}^{n}_{1})^{2}\big )= O\mathopen {}\mathclose {\left(\frac{x_{n}^{4}}{n^{4}}\right)}.\) 
Therefore \(\frac{n}{x_{n}^{2}} \sum _{k=1}^{n} (X^{n}_{k})^{2}\)  converges to \((\alpha +1)/\alpha \)  in \(L^{2}\)  under the unconditioned probability, which suffices to conclude.

Stable regime with a large negative deviation
We treat in this subsection the regime described in Theorem REF , where the limit process is again a Brownian bridge.
Recall the assumptions of this theorem: we assume throughout that \(G^{\prime }(\rho ) < \infty \)  and that there exist \(\alpha \in (1,2]\)  and a slowly varying function \(L\)  such that for every \(0 \leqslant s < \rho \) ,
\(G(s)=G(\rho ) \mathopen {}\mathclose {\left(1-m+\frac{m s}{\rho }+\mathopen {}\mathclose {\left(1-\frac{s}{\rho }\right)}^{\alpha } L\mathopen {}\mathclose {\left(\frac{1}{1-s/\rho }\right)}\right)}.\) 
Recall that in the case \(G^{(2)}(\rho ) < \infty \) , i.e. when \(\alpha =2\)  and \(L\)  admits a finite limit at infinity, say \(\ell \) , the law with generating function \(G(\rho \,\cdot )/G(\rho )\)  has a finite variance, equal to
\(\sigma ^{2} = 2\ell + m - m^{2}.\) 
In this case we set \(r_{n} = \sqrt{n \sigma ^{2}/2}\)  for every \(n\geqslant 1\) , otherwise, we let \((r_{n})_{n \geqslant 1}\)  be a sequence such that \(r_{n}^{\alpha } \sim n L(r_{n})\)  as \(n \rightarrow \infty \) .
Finally, we let
\(x_n = m n + \lambda _{n},\qquad \text{where}\qquad \lambda _{n}/r_{n} \rightarrow -\infty \qquad \text{and}\qquad \lambda _{n}/n \rightarrow 0,\) 
and we write \(b_{n} = \rho (1-\varepsilon _{n})\) , which satisfies \(\Psi (b_{n}) = x_{n}/n\) , so \(\varepsilon _{n} \rightarrow 0\) .
Again, we prove a law of large number for the square of the increments; this completes Theorem REF  and was used in the proof of Theorem REF  in the case \(\lambda = -\infty \) .
Proposition A.4 
The following convergence in probability holds:
\(\frac{\varepsilon _{n}}{|\lambda _{n}|} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\alpha -1 + \frac{m^{2}}{\sigma ^{2}} \mathbb {1}_{\lbrace G^{(2)}(\rho )<\infty \rbrace }.\) 


Again, we proceed exactly as in the proof of Proposition REF : First,
it is sufficient to work under the unconditional law of the random walk \(\widehat{S}^{n}\) , and then conclude by absolute continuity and time-reversal that the same holds for the bridge, which has the same law as \(S^{n}\) . We therefore replace the \(X^{n}_{k}\) 's by a sequence \((\widehat{X}^{n}_{k})_{k \geqslant 1}\)  of i.i.d. copies of a random variable \(\widehat{X}^{n}\)  with generating function \(\widehat{G}^{n} = G(b_{n} \,\cdot ) / G(b_{n})\) .
Second, we rely on the moments of \((\widehat{X}^{n})^{2}\)  calculated in (REF ).
Note that the generating function \(\widehat{G}^{n}\)  satisfies the same assumption that \(G\)  in Sec. REF  and \(\widehat{X}^{n}\)  has the law of \(\xi ^{(b_{n})}\)  there.
Let us first treat the case where either \(\alpha < 2\)  or \(L\)  converges to \(\infty \) . In this case, combining (REF ) and (REF ) we obtain
\(\frac{n \varepsilon _{n}}{|\lambda _{n}|} \mathbb {E}\big [(\widehat{X}^{n}_{1})^{2}\big ]\mathop {\longrightarrow }^{}_{n \rightarrow \infty }\alpha -1.\) 
By the same argument, relying on the expression of the moments from (REF ) and the derivatives of \(G\)  from (REF ) we infer that as \(n\rightarrow \infty \) ,
\(\frac{n \varepsilon _{n}}{|\lambda _{n}|} \mathbb {E}\big [(\widehat{X}^{n}_{1})^{4}\big ]\sim \frac{n \varepsilon _{n}}{|\lambda _{n}|} \frac{b_{n}^{4} G^{(4)}(b_{n})}{G(b_{n})}\sim \frac{(\alpha -1)(\alpha -2)(\alpha -3)}{\varepsilon _{n}^{2}}.\) 
Recall also that we deduced just below (REF ) that \(r_{n} \varepsilon _{n} \rightarrow \infty \) , in addition to \(|\lambda _{n}| \rightarrow \infty \) , hence
\(n \mathopen {}\mathclose {\left(\frac{\varepsilon _{n}}{|\lambda _{n}|}\right)}^{2} \mathbb {E}\big [(\widehat{X}^{n}_{1})^{4}\big ] \mathop {\longrightarrow }^{}_{n \rightarrow \infty }0.\) 
We conclude in this case that \(\frac{\varepsilon _{n}}{|\lambda _{n}|} \sum _{k=1}^{n} (\widehat{X}^{n}_{k})^{2}\)  converges to \(\alpha -1\)  in \(L^{2}\)  under the unconditioned probability, which suffices for our claim.
It remains to treat the case where \(\alpha =2\)  and \(L\)  admits a finite and positive limit, say \(\ell \) . Here we use (REF ) instead.
Moreover in this case \(G^{(2)}(\rho ) < \infty \) ; let \(\widehat{X}\)  have the law with generating function \(G(\rho \,\cdot )/G(\rho )\) , so \(\mathbb {E}[\widehat{X}] = m\)  and let \(s^{2} = \mathbb {E}[(\widehat{X})^{2}]\) , which is given by (REF ), and notice that since \(2\ell = \mathbb {E}[\widehat{X}(\widehat{X}-1)]\) , then \(s^{2} = 2\ell + m\) .
Similarly let \(s^{2}_{n} = \mathbb {E}[(\widehat{X}^{n}_{1})^{2}]\) , then by continuity \(s^{2}_{n} \rightarrow s^{2}\) .
The claim can be proved very much as in the proof of Proposition REF . Recall from there that all we need to prove is that for any \(\varepsilon > 0\) ,
\(\mathbb {E}\Big [\big |(\widehat{X}^{n}_{1})^{2} - s_{n}^{2}\big | \mathbb {1}_{\lbrace |(\widehat{X}^{n}_{1})^{2} - s_{n}^{2}| > n \varepsilon ^{3}\rbrace }\Big ] \mathop {\longrightarrow }^{}_{n \rightarrow \infty }0.\) 
However here we cannot rely on a uniform bound for the fourth moment, and we argue instead by absolute continuity.
Indeed, for every \(k \geqslant 0\) , we have
\(\mathbb {P}(\widehat{X} = k) = \frac{\rho ^{k} p(k)}{G(\rho )}\qquad \text{and}\qquad \mathbb {P}(\widehat{X}^{n}_{1} = k) = \frac{b_{n}^{k} p(k)}{G(b_{n})}.\) 
Consequently, since \(b_{n} \leqslant \rho \)  for every \(n\)  and \(G(b_{n}) \rightarrow G(\rho )\) , then

E[|(Xn1)2 - sn2| 1{|(Xn1)2 - sn2| > n 3}]
= G()G(bn) E[|(X)2 - sn2| 1{|(X)2 - sn2| > n 3} (bn)k]
G()G(bn) E[|(X)2 - sn2| 1{|(X)2 - sn2| > n 3}],

which tends to 0. We conclude as in the proof of Proposition REF  that \(\frac{1}{n} \sum _{k=1}^{n} (\widehat{X}^{n}_{k})^{2}\)  converges to \(s^{2}\)  in probability.
Our claim then follows from (REF ) together with the fact that the constant there equals \(s^{2} - m^{2}\) .
Notice that when \(L\)  has a finite limit at infinity, say \(\ell \) , then \(|\lambda _{n}| / \varepsilon _{n} \sim (2\ell + m - m^{2}) n = (s^{2} - m^{2}) n\)  by (REF )
so \(n^{-1} \sum _{k=1}^{n} (X^{n}_{k})^{2}\)  converges in probability to \(s^{2} = \mathbb {E}[\widehat{X}^{2}]\) , which is consistent with Proposition REF .

Gaussian regime with a small deviation
We finally treat in this subsection the regime described in Theorem REF , in the case \(\alpha =2\) , and one last time, we prove a law of large number for the square of the increments; this was used in the proof of Theorem REF  in the case \(\alpha =2\) .
We work under the same assumptions on \(G\)  as in the previous subsection with \(\alpha =2\) , that is
\(G(s)=G(\rho ) \bigg (1-m+\frac{m s}{\rho }+\mathopen {}\mathclose {\left(1-\frac{s}{\rho }\right)}^{2} L\mathopen {}\mathclose {\left(\frac{1}{1-s/\rho }\right)}\bigg ),\) 
with \(r_{n}\)  defined as in Sec. REF .
However here we assume that there exists \(\lambda \in \mathbb {R}\)  such that
\(x_n = m n + \lambda _{n},\qquad \text{where}\qquad \lambda _{n}/r_{n} \rightarrow \lambda .\) 
As opposed to the preceding regimes, we shall use a fixed distribution for the equivalent random walk. Precisely, we let \(\widehat{X}\)  have the law with generating function \(\widehat{G} = G(\rho \,\cdot )/G(\rho )\) , so its moments are given by (REF ) with \(b = \rho \) . We let \(\widehat{S}\)  denote a random walk with such i.i.d. steps, so Lemma REF  shows that the path \((\widehat{S}_{i} ; 0 \leqslant i \geqslant n)\)  under \(\mathbb {P}(\,\cdot \mid \widehat{S}_{n} = x_{n})\)  has the same law as our original path \(S^{n}\) .
Proposition A.5 
When \(\alpha =2\) , the following convergence in probability holds:
\(r_{n}^{-2} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }2 + \frac{2m^{2}}{\sigma ^{2}} \mathbb {1}_{\lbrace G^{(2)}(\rho ) < \infty \rbrace }.\) 


As now usual,
it is sufficient to work under the unconditional law of the random walk \(\widehat{S}\) , and then conclude by absolute continuity and time-reversal that the same holds for the bridge, which has the same law as \(S^{n}\) . We therefore replace the \(X^{n}_{k}\) 's by a sequence \((\widehat{X}_{k})_{k \geqslant 1}\)  of i.i.d. random variables with generating function \(\widehat{G}\) .
The latter belong to the domain of attraction of a Gaussian distribution;
in the case where \(L\)  has a finite limit, we have \(\widehat{G}^{(2)}(\rho ) < \infty \)  so the \((\widehat{X}_{k})^{2}\) 's have finite mean and the claim simply follows from the law of large numbers combined with the fact that \(r_{n} = \sqrt{n \sigma ^{2}/2}\)  :
\(n^{-1} \sum _{k=1}^{n} (X^{n}_{k})^{2}\mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\mathbb {E}[\widehat{X}^{2}]=\sigma ^{2}+m^{2}\) 
In particular, the convergence of Proposition REF  also holds for \(b=\rho \)  when \(G^{(2)}(\rho ) < \infty \) .
This can be extended to the case \(\widehat{G}^{(2)}(\rho ) =\infty \)  by looking at the first two truncated moments, as in e.g. [3]}. We reproduce the argument for the reader's convenience. Let \(L_{1}\)  be the slowly varying function defined by \(L_{1}(x) = \mathbb {E}[\widehat{X}_{1}^{2} \mathbb {1}_{\lbrace |\widehat{X}_{1}| \leqslant x\rbrace }]\)  for every \(x > 0\) .
As in e.g. [16]} we have \((L_{1}(x)-m) \sim 2 L(x)\)  as \(x \rightarrow \infty \) . Further, by [17]}, the ratio \(x^2 \mathbb {P}(\widehat{X}_{1} \geqslant x) / L_{1}(x)\)  converges to \((2-\alpha )/2 = 0\)  as \(x \rightarrow \infty \) .
Then for every \(\varepsilon > 0\) , since \(L_1\)  is slowly varying, then \(L_1(\varepsilon r_n) \sim L_1(r_n)\)  and so
\(\mathbb {P}\mathopen {}\mathclose {\left(r_n^{-1} \max _{1 \leqslant i \leqslant n} \widehat{X}_i \geqslant \varepsilon \right)}\leqslant n \mathbb {P}\mathopen {}\mathclose {\left(\overline{X}_{1} \geqslant \varepsilon r_n\right)}= o\mathopen {}\mathclose {\left(n (\varepsilon r_n)^{-2} L_{1}(\varepsilon r_n)\right)}= o\mathopen {}\mathclose {\left(n r_n^{-2} L_{1}(r_n)\right)}= o(1),\) 
where we used that \(n r_n^{-2} \sim L(r_n)^{-1}\)  and that the ratio \(L_{1}(r_n)/L(r_n)\)  is bounded.
Further,
\(\mathbb {E}\bigg [r_{n}^{-2} \sum _{1 \leqslant i \leqslant n} \widehat{X}_{i}^2 \mathbb {1}_{\lbrace |\widehat{X}_{i}| \leqslant \varepsilon r_{n}\rbrace }\bigg ]= n r_{n}^{-2} \mathbb {E}\mathopen {}\mathclose {\left[\widehat{X}_{1}^2 \mathbb {1}_{\lbrace |\widehat{X}_{1}| \leqslant \varepsilon r_{n}\rbrace }\right]}\sim n r_{n}^{-2} L_{1}(r_{n})\sim \frac{L_{1}(r_{n})}{L(r_{n})},\) 
and, similarly, the variance of this quantity equals
\(n r_{n}^{-4} \mathrm {Var}\mathopen {}\mathclose {\left(\widehat{X}_{1}^2 \mathbb {1}_{\lbrace |\widehat{X}_{1}| \leqslant \varepsilon r_{n}\rbrace }\right)}\leqslant \varepsilon ^2 n r_{n}^{-2} \mathbb {E}\mathopen {}\mathclose {\left[\widehat{X}_{1}^2 \mathbb {1}_{\lbrace |\widehat{X}_{1}| \leqslant \varepsilon r_{n}\rbrace }\right]}\sim \varepsilon ^2 \frac{L_{1}(r_{n})}{L(r_{n})}.\) 
Since we have previously shown that with high probability, \(|\widehat{X}_{i}| \leqslant \varepsilon r_n\)  for every \(i \leqslant n\) , then we conclude that \(r_n^{-2} \sum _{1 \leqslant i \leqslant n} \widehat{X}_{i}^2\)  converges in probability to the limit of the ratio \(L_{1}/L\) , which is 2. The completes the proof.
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